Lesson Study in Mathematics: Its

potential for educational improvement in

mathematics and for fostering deep

professional learning by teachers

Professor Max Stephens

Graduate School of Education

The University of Melbourne, Australia

Lesson Study in Japan

• Lesson study needs to be viewed as a

feature teacher professional learning

across the whole-school

• It needs to be supported at all levels of the

school and by educational agencies

beyond the school

• It has a direct relationship to the National

Course of Study

Lesson Study in Japan

• Lesson study is a proving ground for all

teachers

• Lesson Study is about building teacher

capacity – in the long-term

• It is not a hobby for a few teachers, or an

optional extra

• Its focus is on the improvement of

teaching and learning

4

Lesson Study:

A Handbook of Teacher-

Led Instructional

Change

Catherine Lewis (2002)

Research for

Better Schools

5

“Ideas for Establishing

Lesson Study

Communities”

Takahashi & Yoshida

Teaching Children

Mathematics, May, 2004

(NCTM)

6

Lesson Study: A

Japanese Approach to

Improving

Mathematics Teaching

and Learning

Fernandez & Yoshida

(2004)

Lawrence Erlbaum

Associates, Publishers

7

Lesson Study Cycle (Lewis (2002)

1.Goal-Setting

and Planning

2.Research Lesson

3.Lesson

Discussion

4.Consolidation of

Learning

Post Lesson

DiscussionLesson Plan

Lesson Observation

Lesson Study Cycle

• Lesson study is not just about improving a single

lesson

• It is about building pathways for improvement of

instruction

• It contributes to a culture of teacher-initiated

research and to teachers’ collective knowledge

• It focus is always improving children’s

mathematical learning and understanding

(Lewis, 2004, p. 18)

8

Lesson Study Cycle

Planning : making a

detailed lesson plan

How do teachers in

Japan work together

to create a plan for a

research lesson?

9

Goals of

this unit

Related

Units in

previous

and

following

grades

Key items and

questions to ask

Anticipated students’

responses

Teacher’s notes:

how to evaluate

how to use tools,

what to emphasize

Lesson Study Cycle

The Lesson is

a Problem solving

oriented lesson

12

Shulman(1987) on Teacher's knowledge

1) content knowledge

2) general pedagogical knowledge

3) curriculum knowledge

4) pedagogical content knowledge

5) knowledge of learners and their

characteristics

6) knowledge of educational contexts

7) knowledge of educational ends, purposes,

and values, and their philosophical and

historical grounds.

Knowledge for Teaching: three additional

categories

• Knowing how to organize and plan problem

solving oriented lessons.

• Knowing how to evaluate and research teaching

materials

• Knowledge of the lesson study as a continuing

system for building teacher capacity

In lesson study, research on teaching

materials is a key element

• Research on teaching materials involves viewing

the materials with the aim of building Knowledge

for Teaching

• Knowledge for Teaching is knowledge-in-

action

• Knowledge for Teaching requires:

– A mathematical point of view

– An educational point of view

– And from the students’ point of view

15

Understand

Children’s

Mathematics

Explore Possible

Problems, Activities

and Manipulatives

Understand Scope

& Sequence

Understand

Mathematics

Instruction Plan

17

Organization of Japanese Math Lesson

• Presenting the problem for the day

• Problem solving by students

• Comparing and discussing

• Summing up by teacher

Presenting the problem for the day

Stigler & Hiebert (1999) comment that

• “the (Japanese) teacher presents a problem to the

students without first demonstrating how to solve the

problem.”

• “ U.S. teachers almost never do this….the teacher almost

always demonstrates a procedure for solving problems

before assigning them to students.”

• Japanese teachers therefore have to ensure that students

understand the context in which the task is embedded and

the mathematical conditions required for its solution

An example of “Presenting a problem”

• Curriculum-free task: Match Sticks Problems.

• Used in the US–Japan cross cultural research

project (4th and 6th graders) (T.Miwa,1992)

• At that time (1992) ,the task was unfamiliar for

both countries but after that appeared in

textbooks in Japan and it is well known even

internationally

• In Australia it is part of a series of rich

assessment tasks for upper primary and junior

secondary students (Stephens, 2008)

A Mathematically Rich Task

A Mathematically Rich Task

Part A

Do these four strategies give a correct result?

Part B

How many matchsticks would be needed to make 5 cells, 12 cells, 27 cells? Explain your thinking.

Part C

Choose 2 of the above strategies. How do you think the person arrived at his or her strategy? Explain the thinking involved.

23

Number of Matchsticks (Grade 4, 6)

• Squares are made by using matchsticks as shown in the picture. When the number of squares is five, how many matchsticks are used?

(1)Write your way of solution and the answer.

(2)Now make up your own problems like the one above and write them down.

Lesson Study – Grade 4

• In this class, the teacher presented the children with five cells, and asked them to find the number of match-sticks required to make this number of cells

• They were then asked to think about a rule that they could use for this number of cells, and for any other number

• Children developing and explaining their rules are the focus of the lesson

Students work is written on magnetic boards that

are easy to display for the whole class

26

Teacher has carefully selected children’s solutions

for whole class discussion

28

Observers have the teacher’s detailed lesson plan and are

looking at how children and teacher are moving ahead

according to the plan

30

The teacher asked student to explain the

work of another student using geometrical

figures

This student is explaining her visual thinking that

supports her generalisation

32

33

34

35

Why is the teacher highlighting some numbers?

• This was done by the teacher to give emphasis

to the idea that each highlighted number is an

instance of a general pattern – not a number for

calculation.

• She wants the children to see concrete numbers

as generalizable numbers.

• This knowledge-in-action is the result of the

deep research on teaching materials

37

38

This student presents a solution that looks

interesting – but does it generalise?

Here, two versions of the same rule are being compared.

The teacher asks “Which one is easier to follow?”

Teacher is asking students to think about the

visual thinking behind 5×2 + (5+1)

This student explains his visual thinking behind

5×4 – 4, or is it 5×4 – (5 – 1)?

What is the purpose of having children come

to the front and to explain their thinking?

• Sometimes this comparison-discussion activity

may appear to be “show-and-tell”

(Takahashi,2008) but in reality that is not the

case.

• Different student responses have been

anticipated in the lesson plan and are carefully

selected by the teacher to promote deep

mathematical thinking.

44

2×5＋（5+1)＝16

20 – 4＝16

2×10 – 4＝16

3×5＋1＝16

5×2＋4＋2＝16

4×5＝20

20 – 4＝16

4×5 – (5 – 1)＝16

5×3＋1＝16

〈3＋3〉＋2×3＋4＝6＋6＋4＝16

17＋4 – 5＝16

4×3＋4＝16

8×2＝16

5×2＋6

5×2＋（5＋1）＝16

（12 – 4）÷2×2＋8＝16

Some examples of actual students’

work as observed by the teachers

in this research lesson (before

whole class discussion)

Those that contain the red markers

show evidence of generalising (my

red markings)

Post lesson discussion (Professor Fujii is chairing the

meeting, three teachers who taught the lesson are on his

left, all observers are present as is school principal)

At the post-lesson discussion

• Professor Fujii – the external facilitator – introduced the discussion drawing attention to the planning phase and to the goals for these particular lessons – fostering mathematical thinking, visualisation and generalisation

• The principal and her deputy talked about how these lessons meshed in with some over-arching goals of the school – listening and learning from others

– promoting deep thinking

– fostering communication

• Observers, who were other teachers in the school, had been released from regular classes in order to participate in lesson study

• All teachers were expected to attend the discussion which lasted for about 90 minutes

At the post-lesson discussion

• Observers asked teachers about particular points where they had departed from their lesson plan

• Observers asked teachers about specific responses by students

• Teachers brought magnetic boards to refer to and to illustrate particular students’ thinking

• Teachers explained where they thought the lesson had succeeded and where it might be improved next time

Knowledge for Teaching always includes

Mathematical Values

In this lesson, we can note that: Mathematical values are crystallized, such as

• Mathematical thinking needs to be flexible.

• Mathematical expression can also be flexible.

• Seeing concrete numbers as generalizable numbers is important.

• Making a generality visible is important

Knowledge for Teaching always includes

Pedagogical Values

In this lesson, we can note that certain

Classroom culture values are crystallized, such

as

• Moving beyond seeing answers simply as

“wrong” or “correct”

• Listening carefully to friends’ talk

• Express ideas clearly to friends

• Avoid underestimating friends’ ideas

Knowledge for Teaching always includes

Human Values

In this lesson, we can note that certain

Human values are crystallized, such as

• Using previous knowledge and experience

is often needed to solve a new problem

• Learning from errors is important

• In order to clarify A, knowing and being

able to think about non-A is important

51

Sometimes a professor teaches a research

lesson: Why?

Mr Hosomizu’s Grade 5 Lesson

• The lesson we will now see is another

“problem oriented lesson”

• Notice how the lesson follows a similar

format as the one we discussed:– Presenting problem for the day

– Problem solving by students

– Comparing and discussing

– Summing up by teacher

Your thinking about the lesson

• If you had to pick out one or two really

important things mathematical from the

lesson, what would they be?

• Please share your thinking with the person

next to you.

• Are these features what you expect to see

in typical lessons here in Lebanon?

Some comments on the lesson

• Mr Hosomizu’s summation is important: “If we know the result of an expression, we can use it to get the result of another expression”

• Students are expected to deal with mathematical expressions as objects for thinking – not simply as calculations

• These are related to the big ideas of the elementary school curriculum

Some comments on the lesson

• You can work with one problem for a long time provided you don’t focus on the results of the problem but on processes that led to that result

• Students basically used three approaches to simplifying 5.4 ÷ 3

• These are all related to important ideas about equivalence in the elementary school curriculum

Three mathematical procedures

• Enlarge 5.4 to 54, then do 54 ÷ 3, but you

have to remember that when you get an answer

it will be necessary to ÷ by 10

• Change 5.4 ÷ 3 to 54 ÷ 30 in order to get a

result without having to adjust the answer. Some

students did not think this made the problem

easier, but …

• Think of 5.4 as 5.4 metres and so 540 cm, the

convert the answer of 180 cm back to metres

Extending mathematical thinking

• Considering 2.7 ÷ 3, some students repeated

one of the three procedures used for 5.4 ÷ 3

• Mr Hosomizu is happy to accept this, but

• Other students were able to connect this new

problem with the original problem.

• “Knowing the result, and way of calculating, of

an expression is important because we can use

it for other expressions”

Extending mathematical thinking

Finally, students are asked to consider what

other numbers could be used in

÷ 3

where they can use the result of 5.4 ÷ 3 to

find the result of this new expression

Some of the numbers suggested are:

Extending mathematical thinking

• If you know that 5.4 ÷ 3 = 1.8, you can also reason that 2.7 ÷ 3 = 0.9

• Mr Hosomizu asks: If you know these two results, what number can go in the blue box: ÷ 3 such that one of the above results can be used to give the new answer?

• Children suggest: 15.12, 0.35, 410.8, 1.35, 8.1, 3.24, 1.8, 21.6 and 7.1

Extending mathematical thinking

• If you know that 5.4 ÷ 3 = 1.8, you can

also reason that 2.7 ÷ 3 = 0.9

• Children suggest: 15.12, 0.35, 410.8, 1.35,

8.1, 3.24, 1.8, 21.6 and 7.1

• Mr Hosomizu concludes the lesson by

saying that he can understand why

students said 8.1, 1.8, 21.6, 1.35

• To be discussed in the next lesson

For the next lesson

• If you know that 5.4 ÷ 3 = 1.8, you can also reason that 2.7 ÷ 3 = 0.9

• What about 8.1 ÷ 3 = ?1.8 ÷ 3 = ?

21.6 ÷ 3 = ?1.35 ÷ 3 = ?

• “Knowing the result of an expression is important because we can use it for other expressions”

For the next lesson

• If you know that 5.4 ÷ 3 = 1.8, you can also reason that 2.7 ÷ 3 = 0.9

• What about 8.1 ÷ 3 = 2.7 (8.1 = 3 × 2.7)

1.8 ÷ 3 = 0.6 (1.8 = 5.4 ÷ 3)

21.6 ÷ 3 = 7.2 (21.6 = 5.4 × 4)

1.35 ÷ 3 = 0.45 (1.35 = 2.7 ÷ 2)

• “Knowing the result of an expression is important because we can use it for other expressions”

Acknowledgements

• Thanks to Professor Fujii of Tokyo Gakugei University who allowed me to use some parts of his Plenary Lecture at ICME 11 in Monterey Mexico in 2008

• Thanks also to Professor Catherine Lewis from Mills College (Oakland, CA, USA) for previous discussions on the implied values of Lesson Study